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Gyromagnetic Ratio

The ratio of the magnetic
dipole moment to the mechanical angular
momentum of a system.

The electronic spin angular momentum of an electron with spin
s is given by

(1)

where is h-bar,
and in "classical" quantum mechanics, the corresponding magnetic
moment is

(2)

where is the charge on the electron,
is the mass of the electron, and c is
the speed
of light. This gives a value

(3)

where is the Bohr
magneton, sometimes known as the Landé factor. The explanation
of the factor of 2 appearing in this equation, which should
classically be 1 and is caused by orbital electron angular momentum,
was one of the successes of Dirac's theory of the electron
(Griffiths 1987, p. 153).

However, in quantum
electrodynamics, the gyromagnetic ratio has correction terms
given in terms of the fine
structure constant by

(4)

(Hughes and
Kinoshita 1999), where the second term is given erroneously as by Karplus and Kroll (1950;
reproduced in Bethe and Salpeter 1997, p. 93), is the Bohr
magneton, and is h-bar.
The second-order term is known exactly, and is equal to

(5)

(Sommerfeld 1957), where is the Riemann
zeta function and is known as Apéry's
constant. The third-order term is also known exactly, and is given
by

(6)

(Laporta and Remiddi 1996), where is the polylogarithm,
and no simpler form is known for the expression .

In 1947, Julian
Schwinger first determined the first-order correcting term of using quantum
electrodynamics. The original calculation was done by hand, and
can now be done by physics graduate students in about 30 minutes,
with the solution fitting on a single page of an exam notebook. As
laboratory techniques have improved, theorists have made more and
more precise calculations to keep pace with them. In 1957, physicist
Toichiro
Kinoshita computed the second-order correction term after a few
years of hard work using paper and pencil. Kinoshita
subsequently computed a numerical solution for the
third-order term, reporting a crude initial solution in 1972, and
finally a highly accurate value in 1995. The analytic solution for
this term was determined the next year by Laporta and Remiddi
(1996).

Kinoshita
already had more than fifteen years invested at the next
level of accuracy. His first crude solution for the fourth-order
term was proposed in 1981, and its precision improved steadily with
the development of more and more powerful computers. Subsequent
terms in the series become increasingly more difficult to compute,
and involve analysis of thousands of individual Feynman
diagrams. In particular, the fourth-order term is not known
analytically, and requires evaluation of 891 four-loop Feynman
diagrams (Hughes and Kinoshita 1999). Since the gyromagnetic
ratio can be measured to extremely high precision using
spectroscopic techniques, the most accurately known value of is obtained by combining
measurements of with an accurate expansion of the form (6).

The most recent and precise measurements of g involve
observation of microwave-induced transitions between Landau-Rabi
levels of an electron
in a magnetic
field (Van Dyck et al. 1987, Van Dyck 1990). The best
current value for is given by

(7)

implemented as ElectronGFactor
in the Mathematica
add-on package Miscellaneous`PhysicalConstants` (which can
be loaded with the command <<Miscellaneous`).

Fine
Structure Constant

References

Basdevant, J.-L. and Dalibard, J. "Measuring the Electron
Magnetic Moment Anomaly." Ch. 11 in The
Quantum Mechanics Solver: How to Apply Quantum Theory to Modern
Physics. Berlin: Springer-Verlag, pp. 79-81, 2000.

Berestetskii, V. B.; Lifshitz, E. M.; and Ditaevskii,
L. P. Quantum
Electrodynamics, 2nd ed. Oxford, England: Pergamon Press,
pp. 151-158, 1982.

Bethe, H. A. and Salpeter, E. Quantum
Mechanics of One- and Two-Electron Atoms. New York: Plenum,
p. 93, 1977.

Cornell Theory Center Press Release. "Digging at the Roots of
Physics at the Cornell Theory Center." Sep. 18, 1996. http://www.tc.cornell.edu/er/media/1996/kinoshita.release.html.

Griffiths, D. J. Introduction
to Elementary Particles. New York: Wiley, p. 153, 1987.

Hughes, V. W. and Kinoshita, T. "Anomalous g Values
of the Electron and Muon." Rev. Mod. Phys. 71,
S133-139, 1999.

Karplus, R. and Kroll, N. "Fourth-Order Corrections in Quantum
Electrodynamics and the Magnetic Moment of the Electron." Phys.
Rev. 77, 536-549, 1950.

Kinoshita, T. "The Fine Structure Constant." Rept. Prof.
Phys. 59, 1459-1492, 1996.

Laporta, S. and Remiddi, E. "The Analytical Value of the Electron
at Order in QED." Phys. Lett. B 379,
283-291, 1996. http://xxx.lanl.gov/abs/hep-ph/9602417/

Laporta, S. and Remiddi, E. "Progress in the Analytical
Evaluation of the Electron in QED: The Scalar Part of the Triple Cross
Graphs." Phys. Lett. B 356, 390-397, 1995.

Sommerfield, C. M. "Magnetic Dipole Moment of the Electron."
Phys. Rev. Ser. 2 107, 328-329, 1957.

Van Dyck, R. S. Jr.; Schwinberg, P. B.; and Dehmelt,
H. G. "New High Precision Comparison of Electron and Positron
g Factors." Phys. Rev. Lett. 59, 26-29, 1987.

Van Dyck, R. S. Jr. "Anomalous Magnetic Moment of Single
Electrons and Positrons: Experiment." In Quantum
Electrodynamics (Ed. T. Kinoshita). Singapore: World
Scientific, pp. 322-388, 1990.

© Eric W. Weisstein